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by Kenneth Kin Lum

You just did some probability calculations, and don't know if the answers are
correct.  So you write a program to verify the results.  If you have eight dice,
and throw them all at once, what is the probability that there are AT LEAST
three fives?  Try to write a program that find out the "number of desirable
outcomes" / "number of possible outcomes" by iterating through all the possible
outcomes of the dice throw.

It has a verbose mode to see that the program is running correctly (for the case
2 dice, at least 1 five):

C:\rails\depot>ruby dice.rb -v 2 1
1   [1,1]
2   [2,1]
3   [3,1]
4   [4,1]
5   [5,1]  <==
6   [6,1]
7   [1,2]
8   [2,2]
9   [3,2]
10   [4,2]
11   [5,2]  <==
12   [6,2]
13   [1,3]
14   [2,3]
15   [3,3]
16   [4,3]
17   [5,3]  <==
18   [6,3]
19   [1,4]
20   [2,4]
21   [3,4]
22   [4,4]
23   [5,4]  <==
24   [6,4]
25   [1,5]  <==
26   [2,5]  <==
27   [3,5]  <==
28   [4,5]  <==
29   [5,5]  <==
30   [6,5]  <==
31   [1,6]
32   [2,6]
33   [3,6]
34   [4,6]
35   [5,6]  <==
36   [6,6]

Number of desirable outcomes is 11
Number of possible outcomes is 36

Probability is 0.3055555555555556

C:\rails\depot>ruby dice.rb 8 3

Number of desirable outcomes is 226491
Number of possible outcomes is 1679616

Probability is 0.1348468935756745

It also has a "sample mode" to print out the samples every 50,000 times in the
loop:

C:\rails\depot>ruby dice.rb -s 8 3
1   [1,1,1,1,1,1,1,1]
50001   [3,6,3,4,3,1,2,1]
100001   [5,5,6,1,6,1,3,1]
150001   [1,5,3,5,2,2,4,1]
200001   [3,4,6,2,5,2,5,1]
250001   [5,3,3,6,1,3,6,1]
300001   [1,3,6,3,4,3,1,2]
350001   [3,2,3,1,1,4,2,2]
400001   [5,1,6,4,3,4,3,2]
450001   [1,1,3,2,6,4,4,2]
500001   [3,6,5,5,2,5,5,2]  <==
550001   [5,5,2,3,5,5,6,2]  <==
600001   [1,5,5,6,1,6,1,3]
650001   [3,4,2,4,4,6,2,3]
700001   [5,3,5,1,1,1,4,3]
750001   [1,3,2,5,3,1,5,3]
800001   [3,2,5,2,6,1,6,3]
850001   [5,1,2,6,2,2,1,4]
900001   [1,1,5,3,5,2,2,4]
950001   [3,6,1,1,2,3,3,4]
1000001   [5,5,4,4,4,3,4,4]
1050001   [1,5,1,2,1,4,5,4]
1100001   [3,4,4,5,3,4,6,4]
1150001   [5,3,1,3,6,4,1,5]
1200001   [1,3,4,6,2,5,2,5]
1250001   [3,2,1,4,5,5,3,5]  <==
1300001   [5,1,4,1,2,6,4,5]
1350001   [1,1,1,5,4,6,5,5]  <==
1400001   [3,6,3,2,1,1,1,6]
1450001   [5,5,6,5,3,1,2,6]  <==
1500001   [1,5,3,3,6,1,3,6]
1550001   [3,4,6,6,2,2,4,6]
1600001   [5,3,3,4,5,2,5,6]  <==
1650001   [1,3,6,1,2,3,6,6]

Number of desirable outcomes is 226491
Number of possible outcomes is 1679616

Probability is 0.1348468935756745

```